\(\dfrac{1}{4}.\dfrac{2}{6}.\dfrac{3}{8}.\dfrac{4}{10}.\dfrac{5}{12}...\dfrac{30}{62}.\dfrac{31}{64}=2x\)
\(\Rightarrow\dfrac{1}{2.2}.\dfrac{2}{2.3}.\dfrac{3}{2.4}.\dfrac{4}{2.5}.\dfrac{5}{2.6}...\dfrac{30}{2.31}.\dfrac{31}{2.32}\) = 2x
\(\Rightarrow\dfrac{1}{2}\left(\dfrac{1}{2}.\dfrac{2}{3}.\dfrac{3}{4}.\dfrac{4}{5}.\dfrac{5}{6}...\dfrac{30}{31}.\dfrac{31}{32}\right)\) = 2x
\(\Rightarrow\dfrac{1}{2}\left(\dfrac{1.2.3.4.5...30.31}{2.3.4.5...31.32}\right)\) = 2x
\(\Rightarrow\dfrac{1}{2}.\dfrac{1}{32}=2x\)
\(\Rightarrow2x=\dfrac{1}{64}\)
\(\Rightarrow x=\dfrac{\dfrac{1}{64}}{2}=\dfrac{1}{64}.\dfrac{1}{2}=\dfrac{1}{128}\)
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